ISA 444/544: Business Forecasting

class: center, middle, inverse, title-slide

.title[
# ISA 444/544: Business Forecasting
]
.subtitle[
## 15: Stationarity, Differencing and Unit Root Tests
]
.author[
### <br>Fadel M. Megahed, PhD <br><br> Professor <br> Farmer School of Business<br> Miami University<br><br> <a href="https://twitter.com/FadelMegahed"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" /></svg> <span class="citation">@FadelMegahed</span></a> <br> <a href="https://github.com/fmegahed/"><svg viewBox="0 0 496 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" /></svg> fmegahed</a> <br> <a href="mailto:fmegahed@miamioh.edu"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z" /></svg> fmegahed@miamioh.edu</a><br> <a href="https://calendly.com/fmegahed"><svg viewBox="0 0 384 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M202.021 0C122.202 0 70.503 32.703 29.914 91.026c-7.363 10.58-5.093 25.086 5.178 32.874l43.138 32.709c10.373 7.865 25.132 6.026 33.253-4.148 25.049-31.381 43.63-49.449 82.757-49.449 30.764 0 68.816 19.799 68.816 49.631 0 22.552-18.617 34.134-48.993 51.164-35.423 19.86-82.299 44.576-82.299 106.405V320c0 13.255 10.745 24 24 24h72.471c13.255 0 24-10.745 24-24v-5.773c0-42.86 125.268-44.645 125.268-160.627C377.504 66.256 286.902 0 202.021 0zM192 373.459c-38.196 0-69.271 31.075-69.271 69.271 0 38.195 31.075 69.27 69.271 69.27s69.271-31.075 69.271-69.271-31.075-69.27-69.271-69.27z" /></svg> Automated Scheduler for Office Hours</a><br><br>
]
.date[
### Fall 2025
]

---

## Learning Objectives for Today's Class

- Define a **stationary time series**.

- Understand how plotting techniques can help identify non-stationarity.

- Understand how we can **stabilize a non-stationary time series to achieve stationarity** (differencing to stabilize the mean, and log transformation to stabilize the variance).

- Understand how to use **unit root tests** to test for stationarity.

---
class: inverse, center, middle

# What is Stationarity?

---

## Stationarity

.content-box-gray[
**Definition**: If `$\{y_t\}$` is a **stationary time series**, then for all `$s$`, the distribution of `$(y_t,\dots,y_{t+s})$` does not depend on `$t$`.
]

---
count:false

## Stationarity

.content-box-gray[
**Definition**: If `$\{y_t\}$` is a **stationary time series**, then for all `$s$`, the distribution of `$(y_t,\dots,y_{t+s})$` does not depend on `$t$`.
]

- This means that a **stationary time series**:  
  - is roughly horizontal (i.e., no trend)  
  - has a constant variance (i.e., no heteroskedasticity)  
  - no patterns are predictable in the long-term (i.e., no seasonality)

---

## Class Activity: Stationarity or Non-Stationarity?

.panelset[

.panel[.panel-name[Task]

In a **Kahoot**, we will explore various time series data and determine whether they are stationary or non-stationary.

]

.panel[.panel-name[KR]

]

.panel[.panel-name[Δ KR]

]

.panel[.panel-name[Strikes]

]

.panel[.panel-name[House Sales]

]

.panel[.panel-name[Lynx]

<img src="data:image/png;base64,#15_stationarity_files/figure-html/lynx-1.png" alt="A line plot showing the number of lynx trappings in Canada from 1821 to 1934." width="100%" style="display: block; margin: auto;" />
]

.panel[.panel-name[Beer]

<img src="data:image/png;base64,#15_stationarity_files/figure-html/beer-1.png" alt="A line plot showing Australias quarterly beer production." width="100%" style="display: block; margin: auto;" />
]

.panel[.panel-name[Travel]

<img src="data:image/png;base64,#15_stationarity_files/figure-html/air_passengers-1.png" alt="A line plot showing the number of monthly airline passengers from 1949 to 1960." width="100%" style="display: block; margin: auto;" />
]

]

---

## How to Identify Non-Stationarity in the Mean

- **Visual Inspection** of the **time series plot (line-chart)**:  
  - Look for trends in the time series plot.  
  - Look for seasonality in the time series plot.  
  
- Examine the **ACF** of the time series:  
  + A **slow decay** in the ACF plot `$\longrightarrow$` **non-stationary**.  
  + On the other hand, if the ACF plot **drops to zero quickly** `$\longrightarrow$` **likely stationary**.  
  + **Note:** For non-stationary time series, the **value of `$r_1$` is often large and positive**.

---

## Example: The Non-Stationarity in Kroger Stock Price

.left-code[
.font90[

``` python
import pandas as pd
import datetime as dt
import seaborn as sns

path = "../../data/kroger_stock_2024.csv"
kr = (
  pd.read_csv(path)
  .assign(
    ds=lambda x: pd.to_datetime(x['date']),
    m=lambda x: x['ds'].dt.month_name().str[:3],
  )
)

plt.figure(figsize=(8, 5))
ax = sns.lineplot(
  data=kr, x='ds', y='price', color='black'
  )
sns.scatterplot(
  data = kr, x = 'ds', y = 'price', hue = 'm', 
  s=40, ax=ax, palette='Paired'
  )
ax.set(
  title="Kroger's Daily Adjusted Closing Price",
  xlabel="Date",
  ylabel="Adjusted Daily Closing Price",
)
```
]
]

.right-plot[

<img src="data:image/png;base64,#15_stationarity_files/figure-html/kroger1_out-2.png" alt="Kroger's Stock Price Last Year" width="100%" style="display: block; margin: auto;" />
]

---

## Example: The Non-Stationarity in Kroger Stock Price

.left-code[
.font90[

``` python
from statsmodels.tsa.stattools import acf
from statsmodels.graphics.tsaplots import plot_acf

# Using the data from Slide 7
acf_df = pd.DataFrame(
  {'lag': range(0, 21),
   'acf': acf(kr['price'], nlags=20)
  }
)

# Print the first few rows of the ACF df
acf_df.head()

# Plot the ACF and beautify the plot
plt.figure( figsize=(8, 5) )
plot_acf(  kr['price'], lags=20 )
plt.xticks( ticks=range(0, 21) )
plt.xlabel('Lag')
plt.ylabel('ACF')
plt.title("ACF of Kroger's Stock Price")
plt.show()
```
]
]

.right-plot[
.font80[

```
##    lag     acf
## 0    0  1.0000
## 1    1  0.9737
## 2    2  0.9467
## 3    3  0.9189
## 4    4  0.8954
```

<img src="data:image/png;base64,#15_stationarity_files/figure-html/kroger2_out-4.png" alt="ACF of Kroger's Stock Price" width="100%" style="display: block; margin: auto;" /><img src="data:image/png;base64,#15_stationarity_files/figure-html/kroger2_out-5.png" alt="ACF of Kroger's Stock Price" width="100%" style="display: block; margin: auto;" />
]
]

---

## How Does the ACF of a Stationary Time Series Look?

- For a **stationary time series**, the **ACF** plot will **decay (cut off) quickly** to zero.   
  + Non-significant autocorrelations are within the **blue shaded region**, and are **statistically not different from zero**.

.left-code[
.font90[

``` python
*kr['delta'] = kr['price'].diff()

acf_df = pd.DataFrame(
  {'lag': range(0, 21),
   'acf': acf(
*    kr['delta'], nlags=20, missing='drop'
     ).round(4)
  }
)

acf_df.head()

plt.figure( figsize=(8, 5) )
*plot_acf(kr['delta'], lags=20, missing='drop')
plt.xticks( ticks=range(0, 21) )
plt.xlabel('Lag')
plt.ylabel('ACF')
plt.title("ACF of Kroger's Diff in Stock Price")
```
]
]

.right-plot[

.font80[

```
##    lag     acf
## 0    0  1.0000
## 1    1 -0.0274
## 2    2 -0.0382
## 3    3 -0.1016
## 4    4 -0.0387
```
]

<img src="data:image/png;base64,#15_stationarity_files/figure-html/krog_diff_out-8.png" alt="ACF of Kroger's Change in Stock Price" width="100%" style="display: block; margin: auto;" />
]

---
class: inverse, center, middle

# Mathematical Background on the 95% Limits of the ACF Plot 
# (Understanding how `statsmodels` computes confidence intervals)

---

## 1. Standard Error of ACF for White Noise

For an **uncorrelated white noise** process, the standard error (SE) of the sample ACF at lag $ k $ is:

`$$\text{SE}(\hat{\rho}_k) \approx \frac{1}{\sqrt{N}}$$`

- Here, $ N $ is the **sample size**.
- Assumes the series is **white noise**: no autocorrelation beyond lag 0.
- The **95% confidence interval (CI)** is:

`$$\pm 1.96 \times 
\underbrace{\frac{1}{\sqrt{N}}}_{\color{black}{\text{Standard Error (SE)}}}$$`

💡 **Key idea:** More data `$\longrightarrow$` **Smaller SE** `$\longrightarrow$` **Narrower confidence bands**.

???
**Speaker Notes:**
- The standard error of the ACF is inversely proportional to the square root of $ N $.
- This formula applies under the assumption that the time series is **white noise**.
- The confidence intervals are computed as **$ \pm 1.96 \times SE $**, meaning that the limits shrink when $ N $ is large.
- However, real-world time series often **exhibit autocorrelation**, requiring a more complex SE formula.

---

## 2. Bartlett’s Formula for SE of ACF

For a **stationary** time series, the **SE must account for autocorrelation**:
`$$\text{SE}(\hat{\rho}_k) \approx 
\sqrt{
\frac{1}{N} 
\left( 1 + 2 \sum_{j=1}^{k-1} \hat{\rho}_j^2 \right)
}$$`

The **sum of squared autocorrelations**, `$\underbrace{2 \sum_{j=1}^{k-1} \hat{\rho}_j^2}_{\color{black}{\text{Effect of past lags}}}$`, increases SE, which leads to **wider confidence bands**.

**Hence:**  
-  If the time series is **highly correlated** (e.g., a **random walk**), SE **increases**.

- **Differencing reduces autocorrelation**, making the sum **smaller** and the SE **lower**.

???
**Speaker Notes:**
- In **real-world** time series, past values influence future values, which means autocorrelation exists.
- Bartlett’s formula modifies the white noise SE formula by incorporating **autocorrelation at prior lags**.
- The key takeaway is that **higher autocorrelation increases SE**, which **widens** the confidence intervals.
- If the time series is highly persistent (like a random walk), the CI will be **much wider**.
- Differencing **reduces autocorrelation**, leading to **smaller** SE and **narrower** CIs.

---

## 3. Why Do CIs Widen at Larger Lags?

Even though **autocorrelations shrink at higher lags**, the confidence intervals still widen. .black[.bold[Why?]]

`$$\text{SE}(\hat{\rho}_k) \approx 
\sqrt{
\frac{1}{N} 
\left( 1 + 2 \sum_{j=1}^{k-1} \hat{\rho}_j^2 \right)
}$$`

<br>

**Larger lags** `$\longrightarrow$`  `$(\underbrace{1 + 2 \sum_{j=1}^{k-1} \hat{\rho}_j^2}_{\color{black}{\text{Cumulative sum increases}}})$` `$\longrightarrow$` **Higher SE**.

**Fewer overlapping observations** at higher lags `$\longrightarrow$` **Increased uncertainty**.

<br>

**Result:** Confidence intervals **widen at larger lags**, even if `$\hat{\rho}_k$` is small.

???
**Speaker Notes:**
- Many assume that since $ \hat{\rho}_k $ shrinks at larger lags, the CI should also shrink, but the opposite happens.
- The **cumulative sum** of past autocorrelations **grows** in Bartlett’s formula, leading to **larger SE**.
- Another reason: At higher lags, **fewer pairs of observations** contribute to the estimate, increasing uncertainty.
- This is why **ACF plots often show wider CIs for larger lags**, even if $ \hat{\rho}_k $ is close to zero.

---

## 4. Impact of Differencing on Confidence Limits

When **differencing** a time series:

`$$Y_t = X_t - X_{t-1}$$`

- **Reduces autocorrelation**, shrinking the **Bartlett sum** in SE computation:

`$$\underbrace{\sum_{j=1}^{k-1} \hat{\rho}_j^2}_{\color{black}{\text{smaller after differencing}}}$$`

- **Decreases SE**, leading to **narrower CIs** compared to the original series.
- Explains **why** `plot_acf()` **shows:**
  - .black[.bold[Wider limits for a trending or non-stationary series.]]
  - .black[.bold[Narrower limits after differencing.]]

???
**Speaker Notes:**
- Differencing **transforms** a highly autocorrelated series into a more **stationary** one.
- This has a direct impact on **Bartlett’s formula**: The sum of past autocorrelations **shrinks**, lowering SE.
- Lower SE `$\longrightarrow$` **Narrower confidence intervals**.
- This explains why:
  - A **non-stationary series (random walk)** has **wide CIs**.
  - A **stationary differenced series** has **narrower CIs**.

---

## 🔑 Key Takeaways

✅   Original series with .black[.bold[strong autocorrelation]]  `$\longrightarrow$` **Wider confidence bands**

✅   Differenced series with .black[.bold[weaker autocorrelation]]  `$\longrightarrow$` **Narrower confidence bands**

✅   **CIs widen at larger lags** due to .black[.bold[accumulating estimation uncertainty]]

???
**Speaker Notes:**
- To summarize:
  - If a time series has strong **autocorrelation**, its **confidence intervals are wide**.
  - If we **difference the series**, autocorrelation decreases `$\longrightarrow$` **narrower** CIs.
  - Higher lags **accumulate uncertainty**, which is why CIs expand for larger lags.
- Understanding these statistical effects helps correctly **interpret ACF plots** in time series analysis.

---
class: inverse, center, middle

# Stabilizing Non-Stationary Time Series

---

## Stabilizing the Mean: Differencing

- **Differencing** is a common technique to **stabilize the mean** of a time series.

- The **first difference** of a time series is the **change** between consecutive observations in the original series:  
  - `$\Delta y_t = \underbrace{y'_t}_{\begin{array}{c} \text{First} \\ \text{Difference} \end{array}} = \underbrace{y_t - y_{t-1}}_{\begin{array}{c} \text{Current value} \\ \text{minus previous value} \end{array}}$`  
  
- The **differenced series** will have only `$N-1$` observations since it is not possible to calculate a difference `$y_1'$` for the first observation.

---

## Second-order Differencing

Occasionally the differenced data will not appear stationary and it may be necessary to difference the data a second time:

`$$\begin{align}
\underbrace{y''_{t}}_{\text{Second Difference}} & = \underbrace{y'_{t} - y'_{t - 1}}_{\text{First Difference of the First Difference}} \\
& = \underbrace{(y_t - y_{t-1})}_{\text{First Difference at } t} - \underbrace{(y_{t-1}-y_{t-2})}_{\text{First Difference at } t-1}\\
& = \underbrace{y_t - 2y_{t-1} +y_{t-2}}_{\text{Final expanded form}}.
\end{align}$$`

* `$y_t''$` will have `$T-2$` values.

* In practice, it is unlikely to go beyond second-order differences (a different representation of the **data generation process** may be needed).

---

## Seasonal Differencing

A **seasonal difference** is the difference between an observation and the corresponding observation from the previous year.
`$$y'_t = y_t - y_{t-m}$$`
where `$m=$` number of seasons.

* For monthly data `$m=12$`.  
* For quarterly data `$m=4$`.  
* Seasonally differenced series will have .red[ `$T-m$` ] observations.

---

## Interpretation of Differencing

- **First differences** are the change between **one observation and
the next**.

- **Seasonal differences** are the change between **one observation and the same observation in the previous year** (i.e., year over year changes or changes between one year to the next).

---

## Stabilizing the Variance: Log Transformation

**Log transformation** is a common technique to **stabilize the variance** of a time series.

---

## Stabilizing the Variance and the Mean: Log-Differencing

**Log-differencing** is a combination of **log transformation** and **differencing**.

---

## One Final Step: DeSeasonalizing the Time Series

---

## Order Does Not Matter: Seasonal and First Differences

When both seasonal and first differences are applied `$\dots$`

* it makes no difference which is done first---the result will be the same.

If `$y'_t = y_t - y_{t-12}$` denotes seasonally differenced series, then twice-differenced series is

`$$\begin{align}
y^*_t &= y'_t - y'_{t-1} \\
      &= (y_t - y_{t-12}) - (y_{t-1} - y_{t-13}) \\
      &= y_t - y_{t-1} - y_{t-12} + y_{t-13}\: .
\end{align}$$`

This is equivalent to the first-differenced series of the seasonally differenced series:
`$$\begin{align}
y^*_t &= y'_t - y'_{t-1} \\
      &= (y_t - y_{t-12}) - (y_{t-1} - y_{t-13}) \\
      &= y_t - y_{t-1} - y_{t-12} + y_{t-13}\: .
\end{align}$$`

---

## Why Do Seasonal Differencing First?

Hyndman & Athanasopoulos (2022) recommend applying seasonal differencing first because:

- If seasonal differencing alone makes the series stationary, we don not need further differencing.

- If we first apply regular differencing, the seasonal pattern may remain strong, meaning an extra seasonal differencing step will still be necessary.

- **Seasonal differencing is often more effective** at removing non-stationarity in data with **strong seasonality**.

---
class: inverse, center, middle

# Unit Root Tests

---

## Unit Root Tests: ADF Test

- **Unit root tests** are used to determine whether a time series is **stationary** or **non-stationary**.

- The most common unit root test is the **Augmented Dickey-Fuller (ADF) test**.  
- The null hypothesis of the ADF test is that the time series has a **unit root** (i.e., it is **non-stationary**).

- If the null hypothesis is rejected (e.g., if `$p < 0.05$`), the time series is considered **stationary**.

- The ADF test is available in Python through the `statsmodels` library, and can be accessed using the `adfuller()` function as follows:

.center[
`from statsmodels.tsa.stattools import adfuller`
]

**Note:** See [this link](https://www.statsmodels.org/stable/generated/statsmodels.tsa.stattools.adfuller.html) for more information on the `adfuller()` function.

---

## Unit Root Tests: KPSS Test

- The **Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test** is another unit root test used to determine whether a time series is **stationary** or **non-stationary**.

- The null hypothesis of the KPSS test is that the time series is **stationary** around a **fixed level**.

- If the null hypothesis is rejected (e.g., if `$p < 0.05$`), the time series is considered **non-stationary**.

- The KPSS test is available in Python through the `statsmodels` library, and can be accessed using the `kpss()` function as follows:

.center[
`from statsmodels.tsa.stattools import kpss`
]

**Note:** See [this link](https://www.statsmodels.org/stable/generated/statsmodels.tsa.stattools.kpss.html) for more information on the `kpss()` function.

---

## CoreForecast's Short Cut Based on R's `forecast` Package

- In R, the `forecast` package provides a convenient function `ndiffs()` to determine the number of differences required to make a time series stationary.  
- This function uses the **KPSS test** to determine the number of differences required to make the time series stationary.

- In [CoreForecast](https://nixtlaverse.nixtla.io/coreforecast/differences), we can use the `num_diffs(x: ndarray, max_d: int = 1) -> int` function to determine the number of differences required to make a time series stationary. Similarly, we can use `num_seas_diffs(x: ndarray, season_length: int, max_d: int = 1) → int` to determine the number of seasonal differences required to make a time series stationary.

- **Reference:** [CoreForecast Documentation](https://nixtlaverse.nixtla.io/coreforecast/differences).

---

## Demo: Determining the Number of Differences Required

Let us examine the [airpassengers](https://raw.githubusercontent.com/fmegahed/isa444/refs/heads/main/data/airpassengers.csv) dataset. Our goal is to make this dataset stationary by making the appropriate transformations.

---

class: inverse, center, middle

# Recap

---

## Summary of Main Points

By now, you should be able to do the following:

- Define a **stationary time series**.

- Understand how plotting techniques can help identify non-stationarity.

- Understand how we can **stabilize a non-stationary time series to achieve stationarity** (differencing to stabilize the mean, and log transformation to stabilize the variance).

- Understand how to use **unit root tests** to test for stationarity.

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## 📝 Review and Clarification 📝

1. **Class Notes**: Take some time to revisit your class notes for key insights and concepts.
2. **Zoom Recording**: The recording of today's class will be made available on Canvas approximately 3-4 hours after the session ends.
3. **Questions**: Please don't hesitate to ask for clarification on any topics discussed in class. It's crucial not to let questions accumulate.